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Fungrim entry: 2488bb

Jν(r) ⁣(z)=12rk=0r(1)k(rk)Jν+2kr ⁣(z)J^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} J_{\nu + 2 k - r}\!\left(z\right)
Assumptions:νZandzCandrZ0\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
Alternative assumptions:νCandzC{0}andrZ0\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
TeX:
J^{(r)}_{\nu}\!\left(z\right) = \frac{1}{{2}^{r}} \sum_{k=0}^{r} {\left(-1\right)}^{k} {r \choose k} J_{\nu + 2 k - r}\!\left(z\right)

\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("2488bb"),
    Formula(Equal(BesselJ(nu, z, r), Mul(Div(1, Pow(2, r)), Sum(Mul(Mul(Pow(-1, k), Binomial(r, k)), BesselJ(Sub(Add(nu, Mul(2, k)), r), z)), For(k, 0, r))))),
    Variables(nu, z, r),
    Assumptions(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC